Note that genetically engineered changes might result in anti-aging but not immortality.
But let's add freedom from disease in the mix.
People will still die from accidents, murder, suicide, and war.
Everyone should have the treatment.
Having children should then be made illegal unless there is a way to leave the planet, or until there is a way to leave the planet.
Maintaining a birth rate equal to the death rate or some other approrpiate rate could be determined by a planetary wide lottery allowing winners to have children.
There was a good sci-fi program I remember from some 30 years ago that explored the overcrowding issue - the program was called 1.5. Only people with special licenses could have children and the maximum was 1.5 children. Control was mainatined by having a contrceptive in the water supply. Those who were allowed children had a special supply of water.
The story followed a couple who took a vacation in an isolated area and drank from a well, they didn't know about the contrceptive system. The punchline came when they had twins.
Cris